Unsolved problems in mathematics

This article lists some unsolved problems in mathematics. See individual articles for details and sources.

Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six ones yet to be solved are:
* P versus NP
* The Hodge conjecture
* The Riemann hypothesis
* Yang-Mills existence and mass gap
* Navier-Stokes existence and smoothness
* The Birch and Swinnerton-Dyer conjectureOnly the Poincaré conjecture has been solved. The smooth four dimensional Poincaré conjecture is still unsolved. That is, can a four dimensional topological sphere have two or more inequivalent smooth structures?

Other still-unsolved problems

Additive number theory

* Goldbach's conjecture and its weak version
* The values of $g(k)$ and $G(k)$ in Waring's problem
* Collatz conjecture ($3n\; +\; 1$ conjecture)
* Gilbreath's conjecture

Number theory: prime numbers

* Catalan's Mersenne conjecture
* Twin prime conjecture
* infinitely many prime quadruplets
* infinitely many Mersenne primes (Lenstra-Pomerance-Wagstaff conjecture); equivalently, infinitely many even perfect numbers
* infinitely many regular primes, is their density $e^\{-^1!/\_2\}$
* infinitely many Cullen primes
* infinitely many palindromic primes in base 10
* infinitely many Fibonacci primes
* Is every Fermat number composite for $n\; >\; 4$?
* Is 78,557 the lowest Sierpinski number?
* Is 509,203 the lowest Riesel number?
* Fortune's conjecture (that no Fortunate number is composite)
* Polignac's conjecture
* Landau's problems

General number theory

* abc conjecture
* existence of odd perfect numbers
* existence of quasiperfect numbers
* existence of odd weird numbers
* existence of Lychrel numbers
* Proof that 10 is a solitary number
* existence of Taxicab(5, 2, n) for "n">1.
* Brocard's problem: existence of integers, "n","m", such that "n"!+1="m"² other than "n"=4,5,7

Discrete geometry

* Solving the Happy Ending problem for arbitrary $n$
* Finding matching upper and lower bounds for K-sets and halving lines
* The Hirsch conjecture on the lengths of shortest paths in the vertices and edges of a convex polytope

Ramsey theory

* The values of the Ramsey numbers, particularly $R(5,\; 5)$
* The values of the Van der Waerden numbers

General algebra

* Hilbert's sixteenth problem
* Hadamard conjecture
* existence of perfect cuboids
* Existence of quadratic number fields being Euclidean but not norm-Euclidean

Combinatorics

* Number of Magic squares OEIS|id=A006052
* Finding a formula for the probability that two elements chosen at random generate the symmetric group $S\_n$
* Frankl's union-closed sets conjecture that any family of sets closed under unions has an element contained in half or more of the sets
* The Lonely runner conjecture: if $k$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be more than a distance $1/(k+1)$ from each other runner) at some time?

Graph theory

* Erdős-Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
* The Hadwiger conjecture relating coloring to clique minors
* The Ringel-Kotzig conjecture on graceful labeling of trees
* The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
* Deriving a closed-form expression for the percolation threshold values, especially $p\_c$ (square site)
* Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow

Analysis

* Schanuel's conjecture
* Lehmer's conjecture
* Pompeiu problem
* Is $gamma$ (the Euler-Mascheroni constant) irrational?

Group theory

* Is every finitely presented periodic group finite?
* The inverse Galois problem
* For which positive integers "m", "n" is the free Burnside group nowrap|B("m","n") finite? In particular, is nowrap|B(2, 5) finite?

Other

* Generalized star height problem
* Invariant subspace problem
* Modeling black hole mergers
* Problems in Latin squares
* Problems in loop theory and quasigroup theory

Problems solved recently

* Road coloring conjecture (Avraham Trahtman, 2007)
* The Angel problem (Various independent proofs, 2006)
* Stanley-Wilf conjecture (Gabor Tardos and Adam Marcus, 2004)
* Green–Tao theorem (Terence Tao, 2004)
* Poincaré conjecture (Solution by Grigori Perelman in 2002 now confirmed)
* Catalan's conjecture (Preda Mihăilescu, 2002)
* Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, 2001)
* The Langlands program for function fields (Laurent Lafforgue, 1999)
* Taniyama-Shimura conjecture (Wiles, Breuil, Conrad, Diamond, and Taylor, 1999)
* Kepler conjecture (Thomas Hales, 1998)
* Milnor conjecture (Vladimir Voevodsky, 1996)
* Fermat's Last Theorem (Andrew Wiles, 1994)
* Bieberbach conjecture (Louis de Branges, 1985)
* Four color theorem (Appel and Haken, 1977)

ee also

* Hilbert's 23 problems
* Timeline of mathematics

References

* [http://unsolvedproblems.org/ Unsolved Problems in Number Theory, Logic and Cryptography]
* [http://www.claymath.org/millennium/ Clay Institute Millennium Prize]
* [http://mathworld.wolfram.com/UnsolvedProblems.html Unsolved problems page at MathWorld]
* Winkelmann, Jörg, " [http://www.iecn.u-nancy.fr/~winkelma/mirror/unibas/problem.html Some Mathematical Problems] ". 9 March 2006.
* [http://www.geocities.com/ednitou/ List of links to unsolved problems in mathematics, prizes and research.]

Books discussing unsolved problems

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Books discussing recently solved problems

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Resources

* [http://garden.irmacs.sfu.ca Open Problem Garden] The collection of open problems in mathematics build on the principle of user editable ("wiki") site